If `az_(1)+bz_(2)+cz_(3)=0` for complex numbers `z_(1),z_(2),z_(3)` and real numbers a,b,c then `z_(1),z_(2),z_(3)` lie on a

Options

A. straight line
B. circle
C. depends on the choice of a,b,c
D. none of these

Answer

To solve the problem, we start with the equation given:

**Step 1: Write down the equation.**
We have:
\[ az_1 + bz_2 + cz_3 = 0 \]
where \( z_1, z_2, z_3 \) are complex numbers and \( a, b, c \) are real numbers.

**Step 2: Rearrange the equation.**
We can rearrange the equation to express one of the complex numbers in terms of the others:
\[ az_1 = -bz_2 - cz_3 \]
This implies that \( z_1 \) can be expressed as:
\[ z_1 = -\frac{b}{a} z_2 - \frac{c}{a} z_3 \]
(assuming \( a \neq 0 \)).

**Step 3: Interpret the equation geometrically.**
The expression shows that \( z_1 \) is a linear combination of \( z_2 \) and \( z_3 \). This means that \( z_1 \) lies in the same plane as \( z_2 \) and \( z_3 \) in the complex plane.

**Step 4: Analyze the implications.**
Since \( z_1 \) can be expressed as a linear combination of \( z_2 \) and \( z_3 \), the points represented by \( z_1, z_2, z_3 \) can lie on a straight line if \( a, b, c \) are chosen appropriately. However, if \( a, b, c \) are not fixed, we cannot definitively say they lie on a specific geometric figure like a circle or a line.

**Step 5: Conclusion.**
Thus, the relationship between \( z_1, z_2, z_3 \) depends on the choices of \( a, b, c \). Therefore, the answer is:
**Option 3: Depends on the choice of a, b, c.**

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